To factor $x^2 + 4x - 21$, follow these steps:

Step 1: Identify coefficients

The quadratic expression is in the form $ax^2 + bx + c$, where:

• $a = 1$ (coefficient of $x^2$)
• $b = 4$ (coefficient of $x$)
• $c = -21$ (constant term)

Step 2: Find two numbers whose product is $a \cdot c = -21$ and sum is $b = 4$.

• The two numbers are 7 and -3 because: $7 \cdot (-3) = -21 \quad \text{and} \quad 7 + (-3) = 4$

Step 3: Rewrite the middle term using these numbers.

$x^2 + 4x - 21 = x^2 + 7x - 3x - 21$

Step 4: Factor by grouping.

Group the terms in pairs: $(x^2 + 7x) - (3x + 21)$

Factor out the greatest common factors (GCF) from each group: $x(x + 7) - 3(x + 7)$

Step 5: Factor out the common binomial factor.

$(x - 3)(x + 7)$

Final Answer:

$x^2 + 4x - 21 = (x - 3)(x + 7)$

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5 Related Questions:

1. How do you factor quadratic expressions when the leading coefficient is greater than 1?
2. What are the applications of factoring in solving quadratic equations?
3. How do you check if your factorization is correct?
4. Can you explain how to solve quadratic equations using the quadratic formula?
5. How do you graph a quadratic function after factoring it?

Tip: Always verify your factorization by expanding it back to the original expression!